Explanation Given information: The figure shown below shows the free body diagram of the system Figure-(1) Write the expression for the rate change of momentum. H ˙ G = H G x y z + Ω × H G .........(I) Here, the rate of change of angular momentum is H ˙ G , angular momentum with respect to the frame of reference and angular velocity of the xyz frame is Ω . Write the expression for the angular momentum about the centre of mass. H G = I x ω i + I x y ω j + I x z ω k ……. (II) Here, momentum of inertia about x direction is I x , momentum of inertia about x y plane is I x y , momentum of inertia about x z plane is I x z and the angular velocity is x z . Substitute I x ω i + I x y ω j + I x z ω k for H G , ω i for Ω and 0 for H G x y z in Equation (I). H ˙ G = 0 + ω i × ( I x ω i + I x y ω j + I x z ω k ) = I x z ω j + I x y ω k ……. (III) The shaft lies between x z plane hence the value of I x y is 0 . Substitute 0 for I x y in Equation (III). H ˙ G = I x z ω 2 j + ( 0 ) ω 2 k = I x z ω 2 j .........(IV) The magnitude of A and B is Equal and opposite in direction because the center of mass lies on the line A B as it is symmetrical A and B forms a couple. A = − ( B ) .........(V) Write the expression for the force at B . F B = 4 a i × ( B y j + B z k ) = 4 a B y k − 4 a B z j .........(VI) The magnitude of B in the y direction is 0 . Substitute 0 for B y in Equation (VI). F B = 0 − 4 a B z j = − 4 a B z j .........(VII) Here, the magnitude of couple at B is F B the distance between a A to B is 4 a .the magnitude of B in z direction is B z

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ISBN: 9781259977237

Author: BEER

Publisher: MCG

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Chapter 18.2, Problem 18.68P

To determine

The dynamic reaction at A.

The dynamic reaction at point B.

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Chapter 18.2, Problem 18.67P

Chapter 18.2, Problem 18.69P

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A shaft turning at a uniform speed carries two uniform discs A and B of masses 10kg and 8kg respectively. The centres of the mass of the discs are each 2.5mm from the axis of rotation. The radii to the centres of mass are at right angles. The shaft is carried in bearings C and D between A and B such that AC = 0.3m, AD = 0.9m and AB = 1.2m. It is required to make dynamic loading on the bearings equal and a minimum for any given shaft speed by adding a mass at a radius 25mm in a plane E. USING THE METHOD OF DRAWING m*r and m*r*l diagram Determine: The magnitude of the mass in plane E and its angular position relative to the mass in plane A The distance of the plane E from plane A The dynamic loading on each bearing when the mass in plane E has been attached and the shaft rotates at 200 rev/min. For the bearing loads in the opposite direction determine all the unknown values. For the bearing loads in the same direction, show the diagrams and equations only to use for a possible…

A shaft turning at a uniform speed carries two uniform discs A and B of masses 10kg and 8kg respectively. The centres of the mass of the discs are each 2.5mm from the axis of rotation. The radii to the centres of mass are at right angles. The shaft is carried in bearings C and D between A and B such that AC = 0.3m, AD = 0.9m and AB = 1.2m. It is required to make dynamic loading on the bearings equal and a minimum for any given shaft speed by adding a mass at a radius 25mm in a plane E. Determine: (a) The magnitude of the mass in plane E and its angular position relative to the mass in plane A (b) The distance of the plane E from plane A (c) The dynamic loading on each bearing when the mass in plane E has been attached and the shaft rotates at 200 rev/min. For the bearing loads in the opposite direction determine all the unknown values. For the bearing loads in the same direction, show the diagrams and equations only to use for a possible solution.

Chapter 18 Solutions

Vector Mechanics For Engineers

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